Show that a given multivariable function involving sine, cosine, and a polynomial is continuous everywhere.

Vault Of Euclid

Continuity of a Multivariable Trigonometric Function

Functions and graphs warmup

Let \(f(x,y) = \sin(x^2 \cos y)\). Show \(f\) is continuous everywhere.

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What you need first

that polynomials are continuous
that sine and cosine are continuous
that the product of continuous functions is continuous
that the composition of continuous functions is continuous

What this problem tests

Tests the student's ability to recognize elementary functions and apply the theorems regarding the continuity of their products and compositions.

This is a standard introductory problem in multivariable calculus, demonstrating how to build up the continuity of complex functions from simpler known ones.

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